So I'm going to talk about a piece of work that I have just recently published.
As I was saying to a few people earlier, it's describes a slightly different approach to modelling diseases than you'll be learning in this course.
And hopefully I will during this talk, I will sort of introduce you to some of the things that you're going to learn later on.
So if it looks a bit mathematical, just just ignore it and it will have you regard it sort of like a cultural experience.
But but there will be these will be things that you will be learning about, even if not in exactly this sort of way.
So my starting point, I started in the cancer epidemiology unit.
Now, does anybody know who this person is? Yes, it is Ricciardo.
And what was Ricciardo famous for? So he's most famous for having sort of established the link between smoking and lung cancer.
And he's also famous for something else, which is he's got his name associated with paper,
the age distribution of cancer and a multi stage theory of carcinogenesis.
Now going back to the 1950s that they came up with a very, very simple idea for how cancer occurs.
And the thinking was, well, cancer occurs through a sequence of steps, one thing after the other.
So genetic mutations, for example, and until you've had every single one of those mutations,
your your your healthy, but when that final mutation hits, then you know, you've got problems.
And what they realised was that if, if the rates of each of these mutations have a constant rate,
then the distribution of of cancer would be expected to go like age to some power or time some power.
And when they looked at the data and sure enough that that was sort of roughly what they found.
So when you take the log of the that the instance from possible, it was the log of h you see a straight line and that's what they found.
And the slope of that line will tell you the number of stages that are involved in the model.
So that was all seem to be pretty encouraging at the time, but it's not been used very much really since then and the several good reasons for that.
So it's certainly used to be difficult to formulate and understand.
It was certainly difficult to fit to the data and in some cases there are questions about whether the model was identifiable.
So that's whether you could really determine what these parameters were representing.
And perhaps most importantly of all, alternative methods came along, such as logistic regression and cox proportional hazards,
which are actually more useful for most of the things that people were interested in now.
George Box famously said that all models are wrong, but some are useful.
And what he meant by that was that a model can help you to sort of describe and understand processes.
And this is very much the approach I'm taking in the talk today.
And, and in my approach to this work, despite not having been used that much when it has been used,
this model has tended to be associated with important pieces of work.
And I've just listed a handful of examples here and the list needs to be updated.
But what I would argue is that there are many of the reasons for not using this model are no longer present.
So I would argue that it is now comparatively easy to formulate these things and to understand what's going on.
It's easy to fit the data. These issues are identifiable.
They are not really necessarily that important in practise.
And although there are alternative methods, the value of these sort of multi-stage models are sort of they're different.
So so there there is a value to them.
And there's one other thing that really got me interested in them, and this is that they're designed to model sequences of somatic mutations.
And in the last five or six years, one of the really big discoveries in, in my opinion, is the realisation that we are all as we age,
we are cells, accumulate somatic mutations and we get these clones of cells that are mostly harmless but accumulate over time.
So moles in your skin or a very obvious visible example.
So as we age, we, we, we gain these, these mutations in our, you know, in our stem cells.
And most of them are harmless. But by the time we start to get into into old age, it turns out that if you look at your cellular, your blood supply,
that quite a substantial proportion of your blood can be contain these these clones that are distinct from healthy, healthy blood cells.
And it turns out that almost everywhere that people have looked at in your blood and your throat and.
In even in single cells.
This is found to be the case and is also being been found that these these are almost certainly involved in cardiovascular diseases.
My thinking was what other diseases might might these things be involved with?
And is this a reason to look again using multi-stage models to study disease?
Another benefit of these models that distinguishes them from, say,
proportional hazards or logistic regression is that they have what is called a parametric model.
So by that, we specifically model the incidence of disease as a function of age or function of time,
which is different to say logistic regression or proportional hazards,
where either you get the disease or you don't get the disease and you're just estimating how,
say, smoking or or body mass index, for example, modifies your risks.
Now, David Cox, I don't know whether any of you have heard of him.
He unfortunately recently died this year. And in his nineties he devised the proportional hazards and the logistic regression models.
However, in a 1994 interview,
he remarked that he would normally want to tackle problems parametric with his actual specific description for how the incidence of disease occurs,
for example.
And the reason why, he said, was because various people have shown that the answers are very insensitive to the exact formulation of the model.
So in other words, it doesn't really matter if the model is imperfect, it still gets about the right answer.
But you have all the other benefits that goes with having it, which are you tend to it gives you a better understanding of what's going on.
They're often more informative and sort of easy to interpret, and I'll give you examples of that.
And in addition, they allow you to extrapolate. So your data may lie with people aged between 40 and 60,
but you can extrapolate bit forward or back to get an idea of how the incidence will change with age.
But there are very good reasons why most people don't use these sorts of models.
And the first is that you don't need you don't need to go to all this extra effort of describing
the incidence of disease to be able to estimate these the factors that you're interested in.
The, for example, how smoking it modifies your risk of disease.
And in addition to this, for the proportional hazards model, for example,
allows you to model quite complex situations such as where you might have two different groups with very complex varying incidence patterns.
And it would still the analysis would still in principle give you the correct answer.
So if your interest is in estimating associations with, say, smoking and lung cancer,
then a non part model is like parametric model is likely to be better.
And that's no doubt why most of your syllabus is is aimed at just that, and rightly so.
As a result of this, the the age dependent incidence of disease and some of these statistical problems
are needed to to study these things correctly have tended to be ignored.
And as a result, in my opinion, some potentially important patterns of of disease incidence have been overlooked.
And I'm going to show some of those later on.
And I should emphasise that most of the results here you couldn't have got with
your with a conventional logistic regression or proportional has this model.
And what we're going to find we're going to find that a large class of diseases are statistically at least potentially avoidable over a lifetime.
That's what I'm going to argue. And that your effective age at risk of these diseases.
So that's the age you had the equivalent age of somebody with no risk factors to your age.
If, say, your you smoke and you're overweight, for example.
And I'm going to argue that this effect of age risks, these more sort of sporadic diseases,
is more sensitive to risk factors than for diseases that tend to be more later onset.
I'm also going to argue that about 60% of the diseases in the UK Biobank are consistent with this with a multi-stage disease process.
But I'm more but unfortunately,
it also appears that you can't you cannot conclude that the the disease process cells have this multi-stage underlying aetiology.
And I now have evidence for all of this.
I'm going to introduce the effect of age and a relative ageing rate, and a handful of other minor results will crop up along the way.
So this was the seizure of the original multi-stage model of a golf course.
Genesis Imagine disease, cancer in particular, occurring through a sequence of steps.
Now, in its most general form,
a multi stage model of disease is one in which you go from a healthy to a disease state in one of any number of independent paths.
The only important thing is they must be independent of each other.
They can't interact and they through a series of sequential or non sequential states.
So in its most general form you can take any model of that, that type, and you can write down an equation for it.
Don't worry about the equations. That's just to show that they you can do that.
However, in practise, even if you have a situation where this is this is actually how a disease may occur,
you're only going to see the process that happens most rapidly.
So even if a disease can occur in several different ways, if most of the cases are occurring in the same way,
that's that's the sequence of steps you're going to see. And so in practise, the model is going to be a lot simpler.
And when the the hazard of of a disease is small, which is which will turn out to be the case for most diseases,
then you can approximate this, this the situation with what's called a viral distribution.
And I'll explain what that is in just a moment. So if you don't know what these three things are yet, you will do by the end of the course, I presume.
On the left is your survival function. So this is the probability of surviving disease to a certain age.
So. So this is, say, age 100 say, and your probability of surviving today is about half.
Say, for example, the probability density is is or probability distribution is often referred to.
So if you take if you if you look at this interval between 101 and you calculate the end of the curve,
that's the probability of of getting a disease with it within that age range.
Okay. And then the third thing is a little bit less familiar, and he's not taught that often.
And it probably ought to be taught more generally than just in statistics classes.
And this is the hazard function. And what the hazard function is, is it says, okay, if you came to me now and said,
okay, Anthony, what's the chance I'm going to die of lung cancer in the next day?
Well, that's what the hazard function tells you. Okay.
So the hazard function is the probability of getting a disease, given that you haven't got it until now.
Okay. And all three of these things are related.
You can calculate each one in terms of the other. It requires a certain amount of ingenuity.
But you can do that. And one of the thing that most people are unaware of is that for most diseases that occur, if you're otherwise healthy,
you don't have any underlying conditions for almost all diseases that you look at and almost all people,
your risk of a disease up until 80, 90 is very, very low.
It's just the fact there's so many diseases that it makes it it makes them almost inevitable.
And for that for that reason, you can approximate this probability density by by the hazard function,
which makes a lot of the maths far simpler, although you won't see any of that.
So this is actually data from UK Biobank where I've looked at the probability of a disease having occurred by a given age,
given that you haven't yet had any, any underlying diseases. And so this is age 58, 60, 70, 80, 90.
So even by age 100, of course, that age is less good for that.
But but the likelihood of of getting that so this is the along the bottom here, we've got the probability of having had disease by that age.
It's comparatively small. It's really, really surprisingly small.
And I'm not the only person to observe this, but but it's not commonly known.
So it is also cropped up in a statistical genetics paper, an important statistical genetics paper just recently.
Just bear with me with the maths for a little bit longer. As I said, you can relate these three things to each other.
So if you look at the area underneath this, it has a function that gives you what's called the cumulative hazard function.
For those of you who know a bit of calculus, it's the is the integral of it.
And that allows you to express the survival function in terms of the hazard function as the exponential over there.
So where I'm going with this is, is to show you what the viral distribution is.
This is our survival function. And we we write it in terms of the hazard function.
And if we have a proportional hazards model that you can learn about later.
Then what that assumes is you can write the hazard function as a product of a term that's the same
for everybody and that's the has all the age dependents in it and the time dependents in it,
and a factor which is the relative risk. So for example, X might be one if you're a smoker, not if you zero, if you're not.
So it would just be the relative risk, be one if you're a non-smoker and it would be the relative risk be to the beta if you were a smoker.
Okay. And that's the proportional hazards assumption.
And if you are assuming a wider distribution, then all you're doing is assuming that this hazard based based is called baseline.
Hazard function is a power of time or power of all tend to use time and age interchangeably.
And so that's what a Bible distribution is.
Or if you prefer words in a Bible distribution that has a function, it is a power of time or of age, depending on how you express it.
Okay. And as I said, and for diseases where you know that their interest rate is low, which is is most which is already something to look at,
any way you can approximate the wider distribution is a good approximation to this very simple sequential model of disease,
the very original Armitage doll model of cancer.
Okay. So if we plot, if we take the log of this of the both sides,
we should see a straight line if this is the case and we can start to estimate these parameters.
And for roughly 60% of the disease I looked at.
So I've just these are just all I'm trying to do is show that there's lots of these things.
I looked at hundreds of them, literally, and you get a nice straight line and it all looks pretty consistent.
And you can do statistical tests to check that it's consistent with the model.
And and that was turned out.
This is this is very prevalent, however.
So you might say so why is this not being noticed before? And there is a reason for that the data.
So let's say this is your survival your your chance of surviving a disease up to age 100.
Now, most of the data you get will be of people within a certain age.
So for UK Biobank, data will be collected on people, say between roughly age 1460, for example.
What that means is, is you're not seeing you're not seeing this part of the curve.
You're not seeing data from here.
You're just just all you're saying is this now the most advanced, traditional methods for analysing this sort of data?
They will correctly adapt for the fact that this is this is not flat here.
And they will they will get that right. But what they what they don't do because they are not parametric,
they're not modelling the overall age dependence, they are not able to estimate how far this has dropped down.
So how much lower the how much disease would already have occurred in your underlying population?
And if you don't account for that, then instead of seeing this nice straight line, you see something that's very, very curved, very strongly curved.
And so you might just quite naively plot your data and reach the conclusion that this this model fits very, very few cases.
But if you adjust for the fact that your data does not cover the entire human lifetime, then you get a nice a nice straight line again.
And this is why I think this has been overlooked so far.
Okay. I think this is the last in your question. Just ignore this one if you like.
So I thought, well, you know, there are other models you can look at.
So I thought, well, let's also look at a model where disease can occur.
The steps could go in any order. So this one could occur than this one.
Than this one, as opposed to needing to occur in sequence. And I also fit that to the data.
And what I found is that also fit the data pretty well. But it also fit the same types of diseases pretty well.
So my conclusion is, yes, it's conceivable, based on these results,
that many diseases may take may have this sort of multi-stage underlying aetiology or underlying process,
but we can't actually conclude from this data that any of them actually actually do,
because these two models have very different different biological interpretations.
So that's that's work to be ongoing to determine whether it's the case or not.
Now, as I said, there are, however, other advantages to having this nice parametric model that describes the incidence of disease with age.
And one is extrapolation. So you can quantify differences in the patterns of disease onset.
So what I'm showing here is along the bottom is the probability of having experienced a disease by age 50.
Okay, so down that's on the right hand side are diseases that are more likely to be observed by age 50 going up vertically.
Is the relative increase in risk by age 100?
So at the top here, your these diseases, your risk are rapidly increasing with age.
Now, it turns out the what you find is that these overall these are all the disease and they're sort of spread along the diagonal broadly here.
So down here, these diseases are, you know, there's a good chance you might get them in middle age,
but your risk increases comparatively slowly as you age.
In contrast, these ones, you're very, very unlikely to get until you're quite old.
But the risk rapidly increases. And if you you look at the types of diseases which these are, they are the sort of things you might have guessed.
So these are more, you know, cancers and cardiovascular diseases, and these are your digestive diseases,
musculoskeletal diseases and sort of things of unknown origin or poorly defined.
So this is another way of looking at it. So this is this is the hazard of your disease.
So the problem. So if you live to age 80, it's what's the chance of getting a disease in the next year?
And you can see that for the spread of diseases, you know that there's always a risk throughout your life, but it increases really quite slowly.
Whereas these late onset diseases, you know, you're very unlikely to see them until you're older.
But when you're older, you know they are going to get us.
You know, it's not much we statistically at least it doesn't look good.
So the next question is, well, how how modifiable are risks of these things?
So these are now looking at relative risks for established risk factors.
So there's education, deprivation, BMI, height, drinking, diabetes and smoking.
And what I've done is, is I've done box plots.
So I've I've taken I've made estimates of all these things for all the sporadic diseases and all the late onset diseases.
And the the box plot shows the median and the and the extent of the box gives you an idea of I think it's the upper quartile,
gives you an idea of the spread of of the the estimates and the and the risks,
the relative risks are pretty similar, perhaps a bit higher for diabetes and smoking in the late onset diseases.
Now. I think this is my last equation.
Now, if you go back to Weibel distribution, you can take this relative risk and you can pop it inside the brackets here.
And then what you get is something that's equivalent to an effective age risk of disease.
So if you had no risk factors that would give you your age dependent risk of disease.
If you have some risk factors that increase your risk, that that makes this term like a higher.
So for example, so your gives you an effective age that could be either higher or lower depending on whether you will increase your risk through,
say, smoking or reducing it through some other other methods, such as a drug to lower your blood pressure, for example.
You could alternatively regard this this factor here as like like a relative ageing rate because you just
multiply that by your actual age and it gives you your effective age at risk of any particular disease for your,
your particular set of risk factors.
And when you look at your the difference between sporadic and late onset disease in terms of this relative ageing rate,
then we see a very different pattern where the sporadic diseases are actually quite modifiable in terms of terms of risk.
So not only are these diseases potentially avoidable over your lifetime, but they're also very modifiable in terms of their risks,
in terms of well known risk factors that we can we all have some choice over.
Okay. So there's one last topic. Now, one of the thing that you can do is you can take your data and you can split it up into pieces.
This is referred to as stretch justification. So, for example, you can split up your data by the smokers and the non smokers.
And as we well know now, if we plot this data appropriately, if it's consistent with this model, we should see us see a straight line.
So what we're seeing here, we've got the smokers in red, the non smokers in blue, and I've sort of picked three, three diseases because nice examples.
And the fact that the smokers are sort of higher up, it corresponds to a higher incidence rate.
This is this is observed data with with a with a line fitted through it.
Now you can start to interpret what these what these things mean.
Let's say we've got a disease process where where you're going through a serial
sequence of steps and you increase the rates of all of those processes.
So you create all of these things. Then what you'll see is a rigid displacement of this this line upwards.
Okay. And that's what you would what you would get for a proportional in a proportional hazards model, which you'll be learning about.
So it's a proportional hazard model. This is what you're modelling. You're able to model these things moving up and down.
And this is what was was the found for this is benign neoplasms of the colon, rectum and anus.
Okay. And that's consistent with that sort of picture.
Another thing that you might have half is that you might increase the rate of one of these steps sufficiently rapidly or enough.
Rather, you just don't see it anymore. So instead of seeing three, three stages, you only see the two.
And then what you'll see is your the line would move upwards and the slope would become less, less steep.
Now, I should stress, if you see data like this, then you shouldn't be modelling it with the proportion model.
Your statistical test may not pick that out. But you shouldn't be.
You shouldn't in principle be doing that. So there's one other thing that might happen.
What happens if you, you know, you're smoking or whatever you're doing is modifying the risk of disease by some other some
other process so that you're the dominant disease process now becomes something different.
Well, then you might have a situation where not only are you increasing the rate, but you might have more, more steps on this situation.
You could even see an increase in the slope. And I think that's what I'm hypothesising is happening with the lung cancer,
because it turns out that the type of cancer you get in smokers is probably different to the type of cancer you get in non-smokers.
But this is all this is all kind of this is all hypothesising, really.
But it gives you a very different sort of perspective on what might be going on in terms of disease.
Know, these are the same results for women. You could stratify by diabetes and its results for women again, just like all other results here.
I note for the that diabetic men seem to be at lower risk than the non-diabetic men of pulmonary embolism.
I don't know why. It's a it's a it's an odd result. So these are my my sort of conclusions, the late onset diseases.
They seem to be inevitable. You know, the rate at which the risk of those increases is dramatic once you get into old age.
But these sporadic diseases, at least from a statistical point of view, it looks like the risk is is all despite existing throughout our lives.
It doesn't increase particularly rapidly. And so you might hope that we could either prevent or avoid them throughout our lifetime.
In addition, these diseases are strongly influenced by well known risk factors.
So it would appear that a substantial burden of hospital admissions is in principle avoidable by just changing our lifestyle habits.
And I have also hoped to present a slightly different statistical approach that is, at the moment, largely unexplored.
So the opportunities I see them going forward. So for the methodology.
Risk stratification using effective age is a very intuitive way.
Also for communication, public communication, effective age is it is easier for many people to understand if somebody says,
you know, if you smoke, you know, and you're 50 years old, your equivalent risk to heart disease, to somebody who's sort of 60, 65,
that's much easier to understand than somebody saying you've got a relative risk of 20 or whatever it might be.
You get the modelling is is improved for some things and we can get some new insights into the causes of diseases.
And of course the big opportunity in principle is that it does look like in print,
it does suggest that a large proportion of diseases might be avoidable or premature hospital admissions maybe, maybe avoidable.
But again, we kind of know this already. It's just quantifying it a bit more, a bit more carefully.
So to finish off, what might you realistically hope? Richard Dole said that death in old age is inevitable, but death before old age is not.
And I think I would certainly agree with the first part of that. It does look like death in old age is inevitable.
However, I think it's time to start asking whether perhaps, perhaps ill health is not.
Perhaps we can get right through to a ripe old age, but avoid most of these conditions before finally richer.
Something gets us. So that point, I think I'll stop.